Related papers: On a representation theorem for finitely exchangea…

Consider a real diagonal deterministic matrix $X_n$ of size $n$ with spectral measure converging to a compactly supported probability measure. We perturb this matrix by adding a random finite rank matrix, with delocalized eigenvectors. We…

In this paper, we give an alternative proof of the fact that, when compounding a nonnegative probability distribution, convex ordering between the distributions of the number of summands implies convex ordering between the resulting…

We characterize a value of an observable by a `sum rule' for generally non-commuting observables and a `product rule' when restricted to a maximal commuting subalgebra of observables together with the requirement that the value is unity for…

We show the following generalizations of the de Finetti--Hewitt--Savage theorem: Given an exchangeable sequence of random elements, the sequence is conditionally i.i.d. if and only if each random element admits a regular conditional…

We provide a composite version of Ville's theorem that an event has zero measure if and only if there exists a nonnegative martingale which explodes to infinity when that event occurs. This is a classic result connecting measure-theoretic…

Ensembles of isotropic random matrices are defined by the invariance of the probability measure under the left (and right) multiplication by an arbitrary unitary matrix. We show that the multiplication of large isotropic random matrices is…

Let $(X_n)_{n\in \mathbb{N}}$ be a sequence of i.i.d. random variables with distribution $\mathbb P(X_1=1)=\mathbb P(X_1=-1)=1/2$. Let $F(\sigma)=\sum_{n=1}^\infty X_nn^{-\sigma}$. We prove that the following holds almost surely…

In a previous article Don Bennett and I looked for, found and proposed a game in which the Standard Model Gauge Group $S(U(2) \times U(3))$ gets singled out as the "winner". This "game" means that the by Nature chosen gauge group should be…

Let $X_N$ be an $N\ts N$ random symmetric matrix with independent equidistributed entries. If the law $P$ of the entries has a finite second moment, it was shown by Wigner \cite{wigner} that the empirical distribution of the eigenvalues of…

A random variable Z will be called self-inverse if it has the same distribution as its reciprocal 1/Z. It is shown that if Z is defined as a ratio, X/Y, of two rv's X and Y (with Pr[X=0]=Pr[Y=0]=0), then Z is self-inverse if and only if X…

A strong mode of a probability measure on a normed space $X$ can be defined as a point $u$ such that the mass of the ball centred at $u$ uniformly dominates the mass of all other balls in the small-radius limit. Helin and Burger weakened…

We prove new mixing rate estimates for the random walks on homogeneous spaces determined by a probability distribution on a finite group $G$. We introduce the switched random walk determined by a finite set of probability distributions on…

Let $G$ be a graph. For a subset $X$ of $V(G)$, the switching $\sigma$ of $G$ is the signed graph $G^{\sigma}$ obtained from $G$ by reversing the signs of all edges between $X$ and $V(G)\setminus X$. Let $A(G^{\sigma})$ be the adjacency…

Recently Conlon, Fox, and the author gave a new proof of a relative Szemer\'edi theorem, which was the main novel ingredient in the proof of the celebrated Green-Tao theorem that the primes contain arbitrarily long arithmetic progressions.…

We introduce and carefully study a natural probability measure over the numerical range of a complex matrix $A \in M_n(\C)$. This numerical measure $\mu_A$ can be defined as the law of the random variable $<AX,X> \in \C$ when the vector $X…

We prove a computable version of de Finetti's theorem on exchangeable sequences of real random variables. As a consequence, exchangeable stochastic processes expressed in probabilistic functional programming languages can be automatically…

Finite mixtures are a flexible modeling tool for irregularly shaped densities and samples from heterogeneous populations. When modeling with mixtures using an exchangeable prior on the component features, the component labels are arbitrary…

The notion of probability plays a crucial role in quantum mechanics. It appears in quantum mechanics as the Born rule. In modern mathematics which describes quantum mechanics, however, probability theory means nothing other than measure…

In this paper, we show that the likelihood-ratio measure (a) is invariant with respect to dominating sigma-finite measures, (b) satisfies logical consequences which are not satisfied by standard $p$-values, (c) respects frequentist…

Agents vote to choose a fair mixture of public outcomes; each agent likes or dislikes each outcome. We discuss three outstanding voting rules. The Conditional Utilitarian rule, a variant of the random dictator, is Strategyproof and…